Seminars and Colloquia by Series

Monday, December 17, 2018 - 11:15 , Location: Skiles 005 , Hongyu Cheng , MSRI & Nankai University , Organizer: Jiaqi Yang
In the infinite-dimensional KAM theory, solving the homological equations is the one of the main parts. Generally, the coefficients of the homological equations are constants, by comparing the coefficients of the functions, it is easy to solve these equations. If the coefficients of homological equations depend on the angle variables, we call these equations as the variable coefficients homological equations. In this talk we will talk about how to solve these equations.
Friday, December 7, 2018 - 16:00 , Location: Skiles 006 , Tara Brendle , University of Glasgow , Organizer: Dan Margalit
The Burau representation plays a key role in the classical theory of braid groups. When we let the complex parameter t take the value -1, we obtain a symplectic representation of the braid group known as the integral Burau representation. In this talk we will give a survey of results on braid congruence subgroups, that is, the preimages under the integral Burau representation of principal congruence subgroups of symplectic groups. Along the way, we will see the (perhaps surprising) appearance of braid congruence subgroups in a variety of other contexts, including knot theory, homotopy theory, number theory, and algebraic geometry.
Friday, December 7, 2018 - 15:00 , Location: Skiles 170 , Rafael de la Llave , School of Mathematics , Organizer: Rafael de la Llave
<p>Given a Hamiltonian system, normally hyperbolic invariant manifolds and their stable and unstable manifolds are important landmarks that organize the long term behaviour.</p> <p>When the stable and unstable manifolds of a normally hyperbolic invarriant manifold intersect transversaly, there are homoclinic orbits that converge to the manifold both in the future and in the past. Actually, the orbits are asymptotic both in the future and in the past.</p><p> One can construct approximate orbits of the system by chainging several of these homoclinic excursions.</p> <p>A recent result with M. Gidea and T. M.-Seara shows that if we consider long enough such excursions, there is a true orbit that follows it. This can be considered as an extension of the classical shadowing theorem, that allows to handle some non-hyperbolic directions</p>
Friday, December 7, 2018 - 15:00 , Location: None , None , None , Organizer: Lutz Warnke
Wednesday, December 5, 2018 - 14:30 , Location: Skiles 249 , Farbod Shokrieh , University of Copenhagen , , Organizer: Padmavathi Srinivasan
Classical Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. I will present a generalized version of this theorem for metric graphs. (Joint work with Chenxi Wu.)
Wednesday, December 5, 2018 - 14:00 , Location: Skiles 006 , Agniva Roy , Georgia Tech , Organizer: Sudipta Kolay
<p>The talk will discuss a paper by Gompf and Miyazaki of the same name. This paper introduces the notion of dualisable patterns, a technique which is widely used in knot theory to produce knots with similar properties. The primary objective of the paper is to first find a knot which is not obviously ribbon, and then show that it is. It then goes on to construct a related knot which is not ribbon. The talk will be aimed at trying to unwrap the basic definitions and techniques used in this paper, without going too much into the heavy technical details.</p>
Wednesday, December 5, 2018 - 13:55 , Location: Skiles 005 , Rachel Greenfeld , Bar Ilan University , Organizer: Shahaf Nitzan
A set $\Omega\subset \mathbb{R}^d$ is called spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Back in 1974 B. Fuglede conjectured that spectral sets could be characterized geometrically by their ability to tile the space by translations. Although since then the subject has been extensively studied, the precise connection between spectrality and tiling is still a mystery.>In the talk I will survey the subject and discuss some recent results, joint with Nir Lev, where we focus on the conjecture for convex polytopes.
Monday, December 3, 2018 - 15:00 , Location: Skiles 006 , Bruce Reznick , University of Illinois, Urbana Champaign , Organizer: Greg Blekherman
One variation of the Waring problem is to ask for the shortest non-trivial equations of the form f_1^d + ... + f_r^d = 0, under various conditions on r, d and where f_j is a binary form. In this talk I&#39;ll limit myself to quadratic forms, and show all solutions for r=4 and d=3,4,5. I&#39;ll also give tools for you to find such equations on your own. The talk will touch on topics from algebra, analysis, number theory, combinatorics and algebraic geometry and name-check such notables as Euler, Sylvester and Ramanujan, but be basically self-contained. To whet your appetite: (x^2 + xy - y^2)^3 + (x^2 - xy - y^2)^3 = 2x^6 - 2y^6.
Series: Other Talks
Monday, December 3, 2018 - 15:00 , Location: Howey N110 , Simon Berman , Georgia Tech (Physics) , Organizer: Rafael de la Llave
Thesis defense: Advisors: Turgay Uzer and Cristel Chandre Summary: Thirty years after the demonstration of the production of high laser harmonics through nonlinear laser-gas interaction, high harmonic generation (HHG) is being used to probe molecular dynamics in real time and is realizing its technological potential as a tabletop source of attosecond pulses in the XUV to soft X-ray range. Despite experimental progress, theoretical efforts have been stymied by the excessive computational cost of first-principles simulations and the difficulty of systematically deriving reduced models for the non-perturbative, multiscale interaction of an intense laser pulse with a macroscopic gas of atoms. In this thesis, we investigate first-principles reduced models for HHG using classical mechanics. On the microscopic level, we examine the recollision process---the laser-driven collision of an ionized electron with its parent ion---that drives HHG. Using nonlinear dynamics, we elucidate the indispensable role played by the ionic potential during recollisions in the strong-field limit. On the macroscopic level, we show that the intense laser-gas interaction can be cast as a classical field theory. Borrowing a technique from plasma physics, we systematically derive a hierarchy of reduced Hamiltonian models for the self-consistent interaction between the laser and the atoms during pulse propagation. The reduced models can accommodate either classical or quantum electron dynamics, and in both cases, simulations over experimentally-relevant propagation distances are feasible. We build a classical model based on these simulations which agrees quantitatively with the quantum model for the propagation of the dominant components of the laser field. Subsequently, we use the classical model to trace the coherent buildup of harmonic radiation to its origin in phase space. In a simplified geometry, we show that the anomalously high frequency radiation seen in simulations results from the delicate interplay between electron trapping and higher energy recollisions brought on by propagation effects.
Monday, December 3, 2018 - 14:00 , Location: Skiles 006 , Oleg Lazarev , Columbia , Organizer: John Etnyre
Weinstein cobordisms give a natural relationship on the set of Weinstein domains. Flexible Weinstein domains are minimal with respect to this relationship. In this talk, I will use these minimal domains to construct maximal Weinstein domains: any two high-dimensional Weinstein domains with the same topology are Weinstein subdomains of a maximal Weinstein domain also with the same topology. Using this construction, a wide range of new Weinstein domains can be produced, for example exotic cotangent bundles of spheres containing many different closed exact Lagrangians. On the other hand, I will explain how the same line of ideas can be used to prove restrictions on which categories can arise as the Fukaya categories of certain Weinstein domains.